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Let $J$ denote the image of $J$-homomorphism spectrum and let $j$ denote its connective cover. I am interested in knowing the cohomology of $j$ i.e. $$ [j, HZ/p]_*$$ as a module over Steenrod algebra. Or dually the homology of $j$ as a comodule over the dual Steenrod algebra. Is there a nice description?

A reference will be appreciated as well.

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    $\begingroup$ If I'm not mistaken this is the subject of Watanabe's paper "On the spectrum representing algebraic K-theory for a finite field", see projecteuclid.org/euclid.ojm/1200778530 $\endgroup$ – Christian Nassau Apr 10 '15 at 13:51
  • $\begingroup$ I found this link in Knapp's "Operations and cooperations in Im(J)-theory" which might also be relevant. $\endgroup$ – Christian Nassau Apr 10 '15 at 13:52
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This was answered by Don Davis, 1975 Bol. Soc. Mat. Mex. In modern notation, the answer is (at p=2) $H^* j = (A \oplus \Sigma^7 A)/I$, where $I$ is the ideal generated by $Sq^1 \iota_0$, $Sq^2 \iota_0$, $Sq^4\iota_0$, $Sq^8\iota_0 + Sq^1 \iota_7$, $Sq^7\iota_7$, and $(Sq^{(0,1,1)} + Sq^{(4,2)})\iota_7$, in Milnor basis notation. If you prefer the admissable basis, this last operation on $\iota_7$ is $ Sq^6 Sq^3 Sq^1 + Sq^7 Sq^2 Sq^1 + Sq^7 Sq^3 + Sq^8 Sq^2 + Sq^9 Sq^1 + Sq^{10}$.

I derive this quite explicitly and elementarily, as a sample of how to use some sage code Mike Catanzaro wrote, in a talk I gave at Northwestern, "http://math.wayne.edu/~rrb/papers/jnwuhand.pdf", where I also give more precise citation of Don Davis's paper.

Briefly,

  1. you observe that $\psi^3 - 1$ must induce $Sq^4 : H^* \Sigma^4 ksp \to H^* ko$,

  2. you compute the ker and coker

  3. compute that $Ext^1 = F_2$, and that the split extension would give the wrong homotopy,

  4. compute the non-trivial extension using a cocycle defining it.

There, I also note that the Adams spectral sequence for this is quite clean: a marvelous $d_2$ clears out all the rubbish, followed by a simple sequence of differentials which follow from the Leibniz rule, to give the groups of order the 2-adic valuation of 8j in dimensions 8j-1.

Added: I should say, modulo the conjecture that a certain homomorphism really does induce $d_2$, I show this. I do believe it is correct.

Also, there is an interesting module over A(2) that comes up in the j/2 case. The module exhibits periodicity, $\Omega^4 M \simeq \Sigma^{12}M$, and has Ext exactly equal to the associated graded of the homotopy of j/2, in filtrations 2 and up. It is defined over $A(2)$.

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    $\begingroup$ I am curious about the module $M$. Can you refer me to a more detailed account of $M$ (if it is written down somewhere)? $\endgroup$ – Prasit Apr 12 '15 at 1:44
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    $\begingroup$ See pp. 38-43 of the notes linked above. It is called F there. I give the periodic resolution, not just a presentation. $\endgroup$ – Robert Bruner Apr 12 '15 at 1:51
  • $\begingroup$ This was computed in MR2171809 Angeltveit, Vigleik ; Rognes, John . Hopf algebra structure on topological Hochschild homology. Algebr. Geom. Topol. 5 (2005), 1223--1290. $\endgroup$ – Robert Bruner Aug 6 at 14:55

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